Which "God" did Varys supposedly see and hear? I know that the most obvious answer would be R'Hllor since the medium was fire, but I think that begs a greater question: "Can the "Gods" in GoT manifest themselves physically, and if so why haven't they done so rather than using human mediums?"

Currently it is unknown if any gods exist in GoT at all. We know the supernatural exists (we see wights, "warging" and "shadow babies" and such like), but currently we don't know if any of those can be attributed to gods of any kind.
– Andres F.Aug 9 '13 at 16:08

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Can you clarify which chapter/book or episode the event you're referencing occurred in?
– MemnochAug 9 '13 at 16:09

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I could be completely off, but I think he mentions it when recounting the story of his castration. That's one of Tyrion's POV chapters in the first half of book three, I think.
– Anthony GristAug 9 '13 at 16:12

@AnthonyGrist Ah, I seem to remember. Wasn't the wizard who castrated Varys trying to summon some sort of demon? Maybe that's it, but I can't remember the exact quote.
– Andres F.Aug 9 '13 at 16:14

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@AndresF. - He did indeed see something that he described as a demon, not a god.
– System DownAug 9 '13 at 17:10

4 Answers
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"I watched him burn my manly parts in a brazier. The flames turned blue, and I heard a voice answer his call, though I did not understand the words they spoke."

And later on in the same chapter:

"Yet I still dream of that night, my lord. Not of the sorcerer, nor his blade, nor even the way my manhood shriveled as it burned. I dream of that voice. The voice from the flames. Was it a god, a demon, some conjurer's trick?"

Varys himself doesn't know who or what the owner of that voice was. Even if gods do exist in the world of Westeros (which is still not a foregone conclusion), they are not the only supernatural beings in the series. We've seen:

the Others,

a shadow monster born out of Melisandre,

the ghostly inhabitants of the House of the Undying,

and demonic shadows summoned by Mirri Maz Duur.

So calling what Varys heard that night a God would simply be wild speculation.

If Varys' story is true, and if there was really a supernatural entity (of any kind: god, demon or whatever) answering to the offerings being made in the brazier, it might have been a deity (meaning, an entity worshiped by men and perceived to have divine status) known as the Lady of Spears, since they share the ritual of burning someone's manhood (I don't think that such a ritual could be common practice in other god's rituals or common practice at all in Essos).

This Lady of Spears could be, in turn, just an aspect or a different name for any other of the (real?) gods in the world of GoT, since the true nature of the gods (if any) has not been explained even in George R.R. Martin's detail-rich world.

For the second part (the greater question) only speculation can be made at this point.
For the first question, we will find out in a few days time (episode 1 of season 6 "The Red Woman") that the deity in question is indeed R'hllor.

!SPOILER WARNING! below this point

Israeli actress Ania Bukstein was recently confirmed for the part of "High Priestess Kinvara".

Seeing her in the recent #2 trailer interacting with Tyrion & Varys inside the throne room of the Great Pyramid of Meereen, pretty much confirms this piece of leaked script we got last year from another actress' (Tehmina Sunny) audition for the same part.
And, although this piece of dialogue may eventually have been altered for the final version of the shooting script, the gist cannot but be the same, as it is most essential to the overall storyline.